The overall long-term goal of this proposed research is to develop efficient algorithms which permit comprehensive computer simulation of renal function. It is proposed to continue the research that during the last seven years has led to very useful mathematical models of the medullary counter-flow system and the whole kidney. To make our models more comprehensive and realistic, further improvements are required in our computer algorithms for the solution of the complicated systems of coupled, stiff, and nonlinear differential equations describing the models. It is hoped to increase the accuracy of the algorithms by several orders of magnitude through the use of progressively higher order piecewise polynomial approximations when transforming the model differential equations to their finite difference (or finite element) analogues. The use of parallel and multiple shooting, collocation, Rayleigh-Ritz and Galerkin, Invariant-Imbedding, Chebyshev expansion, and coefficient expansion methods will be investigated. It is intended to achieve significant improvements in the efficiency of the algorithms by partitioning the equations in accordance with the physiological connectivity of the kidney, the application of sparse matrix techniques and the use of quasi-Newton and projection methods to minimize the computer storage and run-times. It is expected that as a result of the successful application of the above techniques the models will have reached a level of sophistication to permit a serious attack on the problem of parameter estimation. For this purpose the use of continuation, smoothing and related techniques is proposed.